Arithmetic on elliptic curves with complex multiplication. II. (English) Zbl 0584.14027

Buhler, Joe P.; Gross, Benedict H.

Invent. Math. 79, 11-29 (1985).

This paper is a continuation of the study of the arithmetic of elliptic curves with complex multiplication by \(\mathbb Q(\sqrt{-p})\) which was initiated by B. H. Gross [“Arithmetic on elliptic curves with complex multiplication”, Lect. Notes Math. 776 (1980; Zbl 0433.14032)]. In general, the discussion concerns elliptic curves defined over the Hilbert class field, \(H\), of \(\mathbb Q(\sqrt{-p})\) with \(j\) invariant that of the ring of integers \({\mathcal O}\). Such curves admit complex multiplication by \({\mathcal O}\), and those which are isogenous over \(H\) to all their Galois conjugates are called \(\mathbb Q\)-curves. Various results on the eigenspaces of certain Selmer groups are obtained, and in chapter III there is a refinement of the Birch and Swinnerton-Dyer conjecture in the case when the \(\mathbb Q\)-curve has finitely many \(H\)-rational points. The final chapter presents and discusses some computations in support of the conjecture.

Reviewer: Lawrence G. Roberts

MSC:

11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G15	Complex multiplication and moduli of abelian varieties
11G05	Elliptic curves over global fields
14H45	Special algebraic curves and curves of low genus

Keywords:

elliptic curves defined over the Hilbert class field; complex multiplication; Birch–Swinnerton-Dyer conjecture; Q-curve

Citations:

Zbl 0433.14032

References:

[1]	Arthaud, N.: On the conjecture of Birch and Swinnerton-Dyer. Composito Math37, 209-232 (1978) · Zbl 0396.12011
[2]	Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. math.39, 223-251 (1977) · Zbl 0359.14009 · doi:10.1007/BF01402975
[3]	Greenberg, R.: Onp-adicL-functions and cyclotomic fields II. Nagoya Math J.67, 139-158 (1977) · Zbl 0373.12007
[4]	Goldstein, C., Schappacher, N.: Series d’Eisenstein et fonctionsL de courbes elliptiques a multiplication complexe. Crelle J327, 184-218 (1981) · Zbl 0456.12007
[5]	Gross, B.H.: Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Mathematics, Vol. 776. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0433.14032
[6]	Gross, B.H.: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication. Conference on Fermat’s Last Thm. pp. 219-236. Boston: Birkhäuser 1982
[7]	Gross, B.H.: Minimal models for elliptic curves with complex multiplication. Composito Math45, 155-164 (1982) · Zbl 0541.14010
[8]	Gross, B.H., Lubin, J.: The Eisenstein descent onJ 0(N). Invent. math. in press (1984)
[9]	Manin, Y.I.: Cyclotomic fields and Modular Curves. Russian Math Surveys.26, (No. 6), 7-78 (1971) · Zbl 0266.14012 · doi:10.1070/RM1971v026n06ABEH001272
[10]	Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math IHES47, 33-186 (1978) · Zbl 0394.14008
[11]	Mazur, B., Wiles, A.: Class fields of abelian extensions ofQ. Invent. math.76, 179-330 (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599
[12]	Ribet, K.: A modular construction of unramifiedp-extensions ofQ(? p ). Invent. math.34, 151-162 (1976) · Zbl 0338.12003 · doi:10.1007/BF01403065
[13]	Rohrlich, D.: On theL-functions of anonical Hecke characters of imaginary quadratic fields. Duke M. J.47, 547-557 (1980) · Zbl 0446.12010 · doi:10.1215/S0012-7094-80-04733-X
[14]	Rubin, K.: Congruences for special values ofL-functions of elliptic curves with complex multiplication. Invent. math.71, 339-364 (1983) · Zbl 0513.14012 · doi:10.1007/BF01389102
[15]	Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields. Conference on Fermat’s Last Theorem, pp. 237-254. Boston: Birkhäuser 1982 · Zbl 0519.14017
[16]	Stark, H.:L-series ats=1 IV. Adv. in Math35, 197-235 (1980) · Zbl 0475.12018 · doi:10.1016/0001-8708(80)90049-3